Michael Harris was in town for a few days, and we chatted about the relationship between my conjectures on completed cohomology groups with Emerton and the recent work of Scholze. The brief summary is that Scholze’s results are not naively strong enough to prove our conjectures in full, even for PEL Shimura varieties. Motivated by this discussion, I want to give two quite explicit challenges concerning the mod-p cohomology of arithmetic locally symmetric spaces. The first I imagine will be very hard — it should already imply a certain vanishing conjecture of Geraghty and myself which has strong consequences. However, the formulation is somewhat different and so might be helpful.

Fix an arithmetic locally symmetric space \(X\) corresponding to a reductive group \(G\) over \(\mathbf{Q}.\) Let \(\ell\) and \(p\) be distinct prime numbers. Consider the completed cohomology groups

$latex \widehat{H}^d(\overline{\mathbf{F}}_{\ell}) = \displaystyle{\lim_{\rightarrow}} H^d(X(K),\overline{\mathbf{F}}_{\ell}),
\qquad \widehat{H}^d(\mathbf{C}) = \displaystyle{\lim_{\rightarrow}} H^d(X(K),\mathbf{C}),$

where we take the completion over all compact open subgroups. The limit has an action of \(G(\mathbf{A})\) for the finite adeles \(\mathbf{A}\), and so, in particular, has an action of \(G(\mathbf{Q}_p)\). What irreducible \(G(\mathbf{Q}_p)\) representations can occur in \(\widehat{H}^d(\overline{\mathbf{F}}_{\ell})\)? Here is a guess:

Conjecture: If the smooth admissible representation \(\pi\) of \(G(\mathbf{Q}_p)\) occurs as an irreducible sub-representation of \(\widehat{H}^i(\overline{\mathbf{F}}_{\ell})\), then there exists an irreducible representation \(\Pi\) of \(G(\mathbf{Q}_p)\) in characteristic zero such that:

1. The Gelfand-Kirillov dimension of \(\Pi\) is at least that of \(\pi\). Equivalently,
   \(\mathrm{dim} \ \Pi^{K(p^n)} \gg \mathrm{dim} \ \pi^{K(p^n)}.\)
2. Let \(\mathrm{rec}(\Pi)\) and \(\mathrm{rec}(\pi)\) be the Weil-Deligne representations associated to \(\Pi\) and \(\pi\) respectively by the classical local Langlands conjecture and the mod-\(\ell\) local Langlands conjecture of Vigneras. Then

   \((\overline{\mathrm{rec}(\Pi)})^{\mathrm{ss}} \simeq (\mathrm{rec}(\pi))^{\mathrm{ss}}.\)

3. The representation \(\Pi\) occurs in \(\widehat{H}^j(\mathbf{C})\) for some \(j \le i\).

Roughly speaking, this conjecture says that the irreducible representations occurring in characteristic \(p\) are no more complicated than those which occur in characteristic zero. One naive way to try prove this conjecture would be to show that any torsion class lifts to characteristic zero, at least virtually. This conjecture is too strong, however, as can be seen by considering K-theoretic torsion classes in stable cohomology — the mod \(3\) torsion class in \(H^3(\mathrm{GL}_N(\mathbf{Z}),\mathbf{F}_3)\) can never lift to characteristic zero for sufficiently large N because the cohomology over \(\mathbf{Q}\) is zero for all congruence sugroups by a theorem of Borel. The conjecture as stated seems very hard.

In a different direction, here is the following challenge to those trying to understand completed cohomology through perfectoid spaces. (I expect one can prove this by other means, but I would like to see a proof using algebraic geometry.)

Problem: Fix an integer \(d\), and let \(X_g\) be the Shimura variety corresponding to the moduli space of polarized abelian varieties of genus \(g\). Prove that, for \(g\) sufficiently large, the completed cohomology group \(\widetilde{H}^{d}(X_g,\mathbf{F}_p)\) is finite over \(\mathbf{F}_p\).

An equivalent formulation of this problem is to show that the only smooth admissible \(\mathrm{GSp}_{2g}(\mathbf{Q}_p)\)-representations \(\pi\) which occur inside \(\widetilde{H}^{d}(X_g,\mathbf{F}_p)\) are one dimensional.